lecture 5 rheology - impacttectonics.org

Rheology

Earth Structure (2nd Edition), 2004

W.W. Norton & Co, New York

Slide show by Ben van der Pluijm

© WW Norton; unless noted otherwise

Lecture 5

© EarthStructure (2nd ed) 29/23/2014

Rheology is .... the study of deformation and flow of materials

Rheology is

Associated concepts:

� Stress

� Strain, strain rate

� Elasticity

� Viscosity

� Failure and friction

� Plasticity

� Brittle behavior

� Ductile behavior


Ideal material behaviours

Elastic behaviour – Hooke’s law

Viscous behaviour – Newtonian fluids

Viscoelasticity - Kelvin and Maxwell bodies

Other ideal rheologies

Definitions

• A material’s response to an imposed stress is called the rheology.

• A mathematical representation of the rheology is called a constitutive law.

Simple, 1D Hooke body (linear elasticity)


General Behavior: The Creep Curve

Generalized strain vs. time curve shows

primary (I) or “transient”, |

secondary (II) or “steady-state”,

and tertiary (III) or “accelerated” creep.

Under continued stress the

material will fail

When stress is removed, the material relaxes,

but permanent strain remains

Shear strain rate:

Strain rate: time interval it takes to

accumulate a certain

amount of strain:


Rheologic models

• Physical models consisting of strings and dash pots, and associated

strain–time, stress–strain, or stress–strain rate curves for (a) elastic, (b) viscous, (c) viscoelastic, (d) elasticoviscous, and (e) general linear behavior.

• A useful way to examine these models is to draw your own strain–time

curves by considering the behavior of the spring and the dash pot

individually, and their interaction.

• An ideal dashpot or damper creates a force proportional to its velocity; it

cannot be displaced instantaneously by a finite force.

Symbols used:

e = elongation, e˙ = strain rate, σ=stress, E = elasticity, η (eta)=viscosity, t = time, el denotes elastic component, vi denotes viscous component.


Rheologic models


a) Elastic or Hookean behavior: rubber band σ = E . e (σs = G. γ)

b) Viscous or Newtonian behavior: water σ = η . eo ; ∫σ dt = σ.t = η . e

FIGURE 5.3 Models of linear rheologies

Symbols used: e = elongation, e˙ = strain rate, σ=stress, E = elasticity, η (eta)=viscosity, t = time, el denotes elastic

component, vi denotes viscous component.


Rheologic models

c) Viscoelastic orKelvin behavior:

water-soaked sponge, memory foam

D) Elastoviscous or

Maxwell behavior: paint

E) General Linear

behavior or

Burgers behavior

handout




Rheologic models (interactive)

DePaor, 2002


Elastic Constants

DePaor, 2002

Elastic constants equate two 2nd order tensors, so 4th order

tensor (81 components;

9 independent)

Interdependent moduli:

E = 2G (1+ υ) = 3K (1-2υ)




Representative moduli and viscosities for rocks

K = σ/∆V; G = σs/γ; η = σ/eo


Maxwell relaxation time, tM

Like silly putty, the mantle is elastic (earthquakes) or viscous (plumes); it can

behave elastically (earthquakes) or viscously (plumes) as function of time

Elastic behavior < tM < viscous behavior

Maxwell relaxation time: tM = η/G

or, dominance of viscosity over rigidity

Stress relaxes to 1/e (=0.37) of original

stress (exponential decay)

Mantle viscosity of 1021 Pa⋅s, rigidity of 1011 Pa (olivine-dominated mantle)

tM = 1010 s] or on order of 1000 years (e.g., glacial rebound).


Maxwell relaxation time, tM

Mantle flow stress: σ = η . Eo = 1021 . 10-14 = 107 = 10MPa

= η/G is also temperature dependent

FIGURE 5.4 The Maxwell

relaxation time, tM, is plotted as a

function of time and temperature.

• The curve is based on

experimentally derived

properties for rocks and their

variability.

• The diagram illustrates that hot

mantle rocks deform as a

viscous medium (fluid),

whereas cooler crustal rocks

tend to deform by failure.

• This first-order relationship fits

many observations reasonably

well, but is incomplete for

crustal deformation.

Linear 139/23/2014

Linear vs. non-linear models

FIGURE 5.6 Nonlinear rheologies: elastic-plastic behavior. (a) A physical

model consisting of a block and a spring; the associated strain–time curve (b) and stress–strain rate curve (c).

General Linear is Non-linear (elastic-plastic behavior)

(linear for comparison)Linear


General linear behavior is modeled by placing the elastico-viscous and viscoelastic models in series.

Elastic strain accumulates at the first application of stress (the elastic

segment of the elasticoviscous model).

Subsequent behavior displays the interaction between the elastico-viscous

and viscoelastic models.

When the stress is removed, the elastic strain is first recovered, followed by

the viscoelastic component. However, some amount of strain (permanentstrain) will remain, even after long time intervals (the viscous component of

the elastico-viscous model).

The creep curve for

general linear closely

mimics the creep curve that is observed in

experiments on natural rocks

© EarthStructure (2nd ed) 159/23/2014 © EarthStructure (2nd ed) 159/23/2014

Flow stresses (strength curves)

FIGURE 5.7 Variation of creep strength with

depth for several rock types using a geothermal

gradient of 10 K/km and a strain rate of 10–14/s.

Rs = rock salt, Gr = granite, Gr(w) = wet granite, Qz = quartzite, Qz(w) = wet quartzite,

Pg = plagioclase-rich rock, QzD = quartz diorite, Db = diabase, Ol = Olivine-rich rock.


Rock experiments

FIGURE 5.8 Schematic diagram of a triaxial compression apparatus and states of stress in

cylindrical specimens in compression and extension tests.

• The values of Pc, Pf, and σ can be varied during the experiments.


P, T, depth relationship

Moho

The dashed line is the adiabatic gradient, which is the increase of temperature with

depth resulting from increasing pressure and the compressibility of silicates.

FIGURE 5.9 Change of temperature (T) and pressure (Pc) with depth.


Pressure Suppresses fracturing , promotes ductility, Increases strength

FIGURE 5.10 Compression stress–strain curves of Solnhofen limestone at

various confining pressures (indicated in MPa) at (a) 25°C and (b) 400°C.


Temperature Suppresses fracturing, promotes ductility, reduces strength

FIGURE 5.12 Compression stress–strain curves of Solnhofen limestone at various

temperatures (indicated in °C) and pressures

0.1 MPa confining pressure

40 MPa confining pressure


Temperature

FIGURE 5.13 The effect

of temperature changes

on the compressive strength of some rocks

and minerals.

The maximum stress that a rock can support

until it flows (called the

yield strength of a material) decreases with

increasing temperature.


Strain rate

FIGURE 5.14 Stress versus

strain curves for extension

experiments in weakly foliated Yule marble for

various constant strain rates

at 500°C.


Strain rate

eo = 10-6/sec is 30% change in 4 days

eo = 10-14/sec is 30% change in 1 million years

FIGURE 5.15 Log stress

versus log strain rate

curves for various temperatures based on

extension experiments in Yule marble.

The heavy lines mark the

range of experimental data; the thinner part of the

curves is extrapolation to

slower strain rates.

A representative geologic

strain rate is indicated by the dashed line.

Small eo reduces strength


Fluid pressure

Pf ~ 1/Pc Peff = Pc - Pf

FIGURE 5.16 Comparing the effect of fluid pressure on the behavior of limestone

(b) varying confining pressure.(a) varying pore-fluid pressure


Pore-Fluid Pressure

• Natural rocks commonly contain a fluid phase that may originate from the depositional history or may be secondary in origin (for example, fluids released from prograde metamorphic reactions).

• In particular, low-grade sedimentary rocks such as sandstone and shale, contain a significant fluid component that will affect their behavior under stress.

• Experiments show that increasing the pore-fluid pressure produces a drop in the sample’s strength and reduces the ductility.

• In other words, rocks are weaker when the pore-fluid pressure is high.


Fluid pressure

FIGURE 5.17 The effect of water content on the

behavior of natural quartz.

Dry and wet refer to low

and high water content, respectively.


SIGNIFICANCE FOR GEOLOGIC CONDITIONS

Low Pc, high Pf, low T, high eo: promotes fracturing

High Pc, low Pf, high T, low eo: promotes flow


Representative stress-strain curves

A elastic behavior followed immediately by failure, representing brittle behavior.

B, small viscous component (permanent strain) present before brittle failure.

C, permanent strain accumulates before material fails, representing transitional behavior between brittle and ductile.

D no elastic component and work softening.

E ideal elastic-plastic behavior, in which permanent strain accumulates at constant stress above yield stress.

F typical behavior of many experiments, displaying component of elastic strain followed by permanent strain that requires increasingly higher stresses to accumulate (work hardening).

Yield stress marks stress at change from elastic (recoverable or nonpermanent strain) to viscous (nonrecoverable or permanent strain) behavior

Failure stress is stress at fracturing.

of brittle and ductile behavior


Linear and non-linear viscous behavior

Linear rheology Non-linear rheology

A = proportionality constantE = activation energy (100-500 kJ/mol)R = gas constant (8.3 J/K.mol)T = temperature (K)n = stress exponent (2-5)

eo = 1/η . σ

The viscosity is

defined by the slope of the

linear viscous line in (a) and the effective viscosity

by the slope of

the tangent to the curve in (b).

Effective viscosity:


Brittle and Ductile Behavior

Brittle behavior: response of a solid material to stress during which rock loses continuity (cohesion). Brittle behavior reflects the occurrence of fracture mechanisms. It occurs only when stresses exceed a critical value, after body has already undergone some elastic and/or plastic behavior.

Ductile behavior: response of a solid material to stress such that the rock appears to flow mesoscopically like a viscous fluid. In a material that has deformed ductilely, strain is distributed, i.e., strain develops without formation of mesoscopicdiscontinuities. Ductile behavior can involve brittle (cataclastic flow) or plastic deformation mechanisms.

FIGURE 5.21 Brittle (a) to brittle-ductile (b, c) to ductile (d) deformation,


Ductile strain: behavior vs. mechanism

FIGURE 5.19 Deformation experiment with two cubes containing marbles (a),

and balls of clay (b)

frictional sliding of undeformed marbles

Ductile strain accumulates by

different deformation mechanisms.

The finite strain is equal in

both cases and the mode of

deformation is distributed (ductile

behavior) on the scale of the block.

However, the mechanism

by which the deformation

occurs is quite different

Plastic distortion of clay balls into ellipsoids.

© EarthStructure (2nd ed) 339/23/2014 © EarthStructure (2nd ed) 339/23/2014

Strength and Competency

Strength is stress that material can

support before failure.

Competency is relative term that

compares resistance of rocks to

flow.

FIGURE 5.20 Rheological stratification of

the lithosphere based on the mechanical

properties of characteristic minerals.

Computed lithospheric strength (i.e., the

differential stress) changes not only as a

function of composition, but also as a

function of depth (i.e., temperature).

Rock competency scale: rock salt < shale < limestone

< greywacke < sandstone < dolomite < schist < marble < quartzite < gneiss < granite < basalt


Strength and Rheologic Stratification

Combining friction and flow laws (dry and wet)

(Fossen 2010)

lecture 5 rheology - impacttectonics.org

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